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Number of nonempty normal patterns contiguously matched by the n-th composition in standard order.
4

%I #10 Jun 30 2020 01:55:23

%S 0,1,1,2,1,2,2,3,1,2,2,4,2,4,4,4,1,2,2,4,2,4,4,6,2,4,4,7,4,7,6,5,1,2,

%T 2,4,2,3,4,6,2,4,3,6,4,6,7,8,2,4,4,7,3,7,6,10,4,7,6,10,6,10,8,6,1,2,2,

%U 4,2,3,4,6,2,4,4,6,4,6,7,8,2,4,4,7,4,6

%N Number of nonempty normal patterns contiguously matched by the n-th composition in standard order.

%C The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

%C We define a (normal) pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation_pattern">Permutation pattern</a>

%H Gus Wiseman, <a href="https://oeis.org/A102726/a102726.txt">Sequences counting and ranking compositions by the patterns they match or avoid.</a>

%H Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vTCPiJVFUXN8IqfLlCXkgP15yrGWeRhFS4ozST5oA4Bl2PYS-XTA3sGsAEXvwW-B0ealpD8qnoxFqN3/pub">Statistics, classes, and transformations of standard compositions</a>

%F a(n) = A335458(n) - 1.

%e The a(n) patterns for n = 32, 80, 133, 290, 305, 329, 436 are:

%e (1) (1) (1) (1) (1) (1) (1)

%e (12) (21) (12) (12) (11) (12)

%e (321) (21) (21) (12) (21)

%e (231) (121) (21) (121)

%e (213) (122) (123)

%e (2131) (221) (212)

%e (2331) (1212)

%e (2123)

%e (12123)

%t stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];

%t mstype[q_]:=q/.Table[Union[q][[i]]->i,{i,Length[Union[q]]}];

%t Table[Length[Union[mstype/@ReplaceList[stc[n],{___,s__,___}:>{s}]]],{n,0,100}]

%Y The version for Heinz numbers of partitions is A335516(n) - 1.

%Y The non-contiguous version is A335454(n) - 1.

%Y The version allowing empty patterns is A335458.

%Y Patterns are counted by A000670 and ranked by A333217.

%Y The n-th composition has A124771(n) distinct consecutive subsequences.

%Y Knapsack compositions are counted by A325676 and ranked by A333223.

%Y The n-th composition has A334299(n) distinct subsequences.

%Y Minimal avoided patterns are counted by A335465.

%Y Patterns matched by prime indices are counted by A335549.

%Y Cf. A034691, A056986, A108917, A124767, A124770, A181796, A269134, A333222, A333224, A335456, A335457.

%K nonn

%O 0,4

%A _Gus Wiseman_, Jun 21 2020